The function wtd.var(x,w) in Hmisc calculates the weighted variance of x where w are the weights. It appears to me that wtd.var(x,w) = var(x) if all of the weights are equal, but this does not appear to be the case. Can someone point out to me where I am going wrong here? Thanks. Tom La Bone [[alternative HTML version deleted]] ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. |
On 2007-June-01 , at 01:03 , Tom La Bone wrote:
> The function wtd.var(x,w) in Hmisc calculates the weighted variance > of x > where w are the weights. It appears to me that wtd.var(x,w) = var > (x) if all > of the weights are equal, but this does not appear to be the case. Can > someone point out to me where I am going wrong here? Thanks. The true formula of weighted variance is this one: http://www.itl.nist.gov/div898/software/dataplot/refman2/ch2/ weighvar.pdf But for computation purposes, wtd.var uses another definition which considers the weights as repeats instead of true weights. However if the weights are normalized (sum to one) to two formulas are equal. If you consider weights as real weights instead of repeats, I would recommend to use this option. With normwt=T, your issue is solved: > a=1:10 > b=a > b[]=2 > b [1] 2 2 2 2 2 2 2 2 2 2 > wtd.var(a,b) [1] 8.68421 # all weights equal 2 <=> there are two repeats of each element of a > var(c(a,a)) [1] 8.68421 > wtd.var(a,b,normwt=T) [1] 9.166667 > var(a) [1] 9.166667 Cheers, JiHO --- http://jo.irisson.free.fr/ ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. |
Thanks. I have another related question:
The equation for weighted variance given in the NIST DataPlot documentation is the usual variance equation with the weights inserted. The weighted variance of the weighted mean is this weighted variance divided by N. There is another approach to calculating the weighted variance of the weighted mean that propagates the uncertainty of each term in the weighted mean (see Data Reduction and Error Analysis for the Physical Sciences by Bevington & Robinson). The two approaches do not give the same answer. Can anyone suggest a reference that discusses the merits of the DataPlot approach versus the Bevington approach? Tom La Bone -----Original Message----- From: jiho [mailto:[hidden email]] Sent: Friday, June 01, 2007 2:17 AM To: [hidden email]; R-help Subject: Re: [R] Problem with Weighted Variance in Hmisc On 2007-June-01 , at 01:03 , Tom La Bone wrote: > The function wtd.var(x,w) in Hmisc calculates the weighted variance > of x > where w are the weights. It appears to me that wtd.var(x,w) = var > (x) if all > of the weights are equal, but this does not appear to be the case. Can > someone point out to me where I am going wrong here? Thanks. The true formula of weighted variance is this one: http://www.itl.nist.gov/div898/software/dataplot/refman2/ch2/ weighvar.pdf But for computation purposes, wtd.var uses another definition which considers the weights as repeats instead of true weights. However if the weights are normalized (sum to one) to two formulas are equal. If you consider weights as real weights instead of repeats, I would recommend to use this option. With normwt=T, your issue is solved: > a=1:10 > b=a > b[]=2 > b [1] 2 2 2 2 2 2 2 2 2 2 > wtd.var(a,b) [1] 8.68421 # all weights equal 2 <=> there are two repeats of each element of a > var(c(a,a)) [1] 8.68421 > wtd.var(a,b,normwt=T) [1] 9.166667 > var(a) [1] 9.166667 Cheers, JiHO --- http://jo.irisson.free.fr/ ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. |
In reply to this post by jiho
jiho wrote:
> On 2007-June-01 , at 01:03 , Tom La Bone wrote: >> The function wtd.var(x,w) in Hmisc calculates the weighted variance >> of x >> where w are the weights. It appears to me that wtd.var(x,w) = var >> (x) if all >> of the weights are equal, but this does not appear to be the case. Can >> someone point out to me where I am going wrong here? Thanks. > > The true formula of weighted variance is this one: > http://www.itl.nist.gov/div898/software/dataplot/refman2/ch2/ > weighvar.pdf > But for computation purposes, wtd.var uses another definition which > considers the weights as repeats instead of true weights. However if > the weights are normalized (sum to one) to two formulas are equal. If > you consider weights as real weights instead of repeats, I would > recommend to use this option. > With normwt=T, your issue is solved: > > > a=1:10 > > b=a > > b[]=2 > > b > [1] 2 2 2 2 2 2 2 2 2 2 > > wtd.var(a,b) > [1] 8.68421 > # all weights equal 2 <=> there are two repeats of each element of a > > var(c(a,a)) > [1] 8.68421 > > wtd.var(a,b,normwt=T) > [1] 9.166667 > > var(a) > [1] 9.166667 > > Cheers, > > JiHO The issue is what is being assumed for N in the denominator of the variance formula, since the unbiased estimator subtracts one. Using normwt=TRUE means you are in effect assuming N is the number of elements in the data vector, ignoring the weights. Frank Harrell > --- > http://jo.irisson.free.fr/ > > ______________________________________________ > [hidden email] mailing list > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide http://www.R-project.org/posting-guide.html > and provide commented, minimal, self-contained, reproducible code. > -- Frank E Harrell Jr Professor and Chair School of Medicine Department of Biostatistics Vanderbilt University ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
Frank Harrell
Department of Biostatistics, Vanderbilt University |
In reply to this post by Tom La Bone-2
On 2007-June-01 , at 13:00 , Tom La Bone wrote:
> The equation for weighted variance given in the NIST DataPlot > documentation > is the usual variance equation with the weights inserted. The > weighted > variance of the weighted mean is this weighted variance divided by N. > > There is another approach to calculating the weighted variance of the > weighted mean that propagates the uncertainty of each term in the > weighted > mean (see Data Reduction and Error Analysis for the Physical > Sciences by > Bevington & Robinson). The two approaches do not give the same > answer. Can > anyone suggest a reference that discusses the merits of the DataPlot > approach versus the Bevington approach? I am no expert but I did a little research on the subject and it seems there is no analytical equivalent to the standard error of the mean in weighted statistics (i.e. there is no standard error/variance of the weighted mean). That's why you find so many formulas: they are all numerical approximations that make more of less sense. If you have access to DcienceDirect there is a paper which compares 3 of those analytical approximations to the variance estimated by bootstrap: @article{Gatz1995AE, Author = {Gatz, Donald F and Smith, Luther}, Date-Added = {2007-05-19 14:15:58 +0200}, Date-Modified = {2007-06-01 14:12:17 +0200}, Filed = {Yes}, Journal = {Atmospheric Environment}, Number = {11}, Pages = {1185--1193}, Rating = {0}, Title = {The standard error of a weighted mean concentration--I. Bootstrapping vs other methods}, Url = {http://www.sciencedirect.com/science/article/B6VH3-3YGV47C-6X/ 2/18b627259a75ff9b765410aaa231e352}, Volume = {29}, Year = {1995}, Abstract = {Concentrations of chemical constituents of precipitation are frequently expressed in terms of the precipitation-weighted mean, which has several desirable properties. Unfortunately, the weighted mean has no analytical analog of the standard error of the arithmetic mean for use in characterizing its statistical uncertainty. Several approximate expressions have been used previously in the literature, but there is no consensus as to which is best. This paper compares three methods from the literature with a standard based on bootstrapping. Comparative calculations were carried out for nine major ions measured at 222 sampling sites in the National Atmospheric Deposition/National Trends Network (NADP/NTN). The ratio variance approximation of Cochran (1977) gave results that were not statistically different from those of bootstrapping, and is suggested as the method of choice for routine computing of the standard error of the weighted mean. The bootstrap method has advantages of its own, including the fact that it is nonparametric, but requires additional effort and computation time.}} The analytical formula which is the closest to the bootstrap is this one: var.wtd.mean.cochran <- function(x,w) # # Computes the variance of a weighted mean following Cochran 1977 definition # { n = length(w) xWbar = wtd.mean(x,w) wbar = mean(w) out = n/((n-1)*sum(w)^2)*(sum((w*x-wbar*xWbar)^2)-2*xWbar*sum((w- wbar)*(w*x-wbar*xWbar))+xWbar^2*sum((w-wbar)^2)) return(out) } It's the one I am retaining. NB: the part two of the paper cited above may also interest you: @article{Gatz1995AEa, Author = {Gatz, Donald F and Smith, Luther}, Date-Added = {2007-05-19 14:15:58 +0200}, Date-Modified = {2007-06-01 12:11:53 +0200}, Filed = {Yes}, Journal = {Atmospheric Environment}, Number = {11}, Pages = {1195--1200}, Title = {The standard error of a weighted mean concentration--II. Estimating confidence intervals}, Url = {http://www.sciencedirect.com/science/article/B6VH3-3YGV47C-6Y/ 2/a187487377ef52b741e3dabdfca97517}, Volume = {29}, Year = {1995}, Abstract = {One motivation for estimating the standard error, SEMw, of a weighted mean concentration, Mw, of an ion in precipitation is to use it to compute a confidence interval for Mw. Typically this is done by multiplying the standard error by a factor that depends on the degree of confidence one wishes to express, on the assumption that the weighted mean has a normal distribution. This paper compares confidence intervals of Mw concentrations of ions in precipitation, as computed using the assumption of a normal distribution, with those estimated from distributions produced by bootstrapping. The hypothesis that Mw was normally distributed was rejected about half the time (at the 5{\%} significance level) in tests involving nine major ions measured at ten diverse sites in the National Atmospheric Deposition Program/National Trends Network (NADP/NTN). Most of these rejections occurred at sites with fewer than 100 samples, in agreement with previous results. Nevertheless, the hypothesis was often rejected at sites with more than 100 samples as well. The maximum error (relative to Mw) in the 95{\%} confidence limits made by assuming a normal distribution of the Mw at the ten sites examined was about 27{\%}. Most such errors were less than 10{\%}, and errors were smaller at sampling sites with > 100 samples than at those with < 100 samples.}} Cheers, JiHO --- http://jo.irisson.free.fr/ ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. |
Wonderful! Thanks for the assistance.
Tom La Bone -----Original Message----- From: jiho [mailto:[hidden email]] Sent: Friday, June 01, 2007 8:17 AM To: [hidden email] Cc: 'R-help' Subject: Re: [R] Problem with Weighted Variance in Hmisc On 2007-June-01 , at 13:00 , Tom La Bone wrote: > The equation for weighted variance given in the NIST DataPlot > documentation > is the usual variance equation with the weights inserted. The > weighted > variance of the weighted mean is this weighted variance divided by N. > > There is another approach to calculating the weighted variance of the > weighted mean that propagates the uncertainty of each term in the > weighted > mean (see Data Reduction and Error Analysis for the Physical > Sciences by > Bevington & Robinson). The two approaches do not give the same > answer. Can > anyone suggest a reference that discusses the merits of the DataPlot > approach versus the Bevington approach? I am no expert but I did a little research on the subject and it seems there is no analytical equivalent to the standard error of the mean in weighted statistics (i.e. there is no standard error/variance of the weighted mean). That's why you find so many formulas: they are all numerical approximations that make more of less sense. If you have access to DcienceDirect there is a paper which compares 3 of those analytical approximations to the variance estimated by bootstrap: @article{Gatz1995AE, Author = {Gatz, Donald F and Smith, Luther}, Date-Added = {2007-05-19 14:15:58 +0200}, Date-Modified = {2007-06-01 14:12:17 +0200}, Filed = {Yes}, Journal = {Atmospheric Environment}, Number = {11}, Pages = {1185--1193}, Rating = {0}, Title = {The standard error of a weighted mean concentration--I. Bootstrapping vs other methods}, Url = {http://www.sciencedirect.com/science/article/B6VH3-3YGV47C-6X/ 2/18b627259a75ff9b765410aaa231e352}, Volume = {29}, Year = {1995}, Abstract = {Concentrations of chemical constituents of precipitation are frequently expressed in terms of the precipitation-weighted mean, which has several desirable properties. Unfortunately, the weighted mean has no analytical analog of the standard error of the arithmetic mean for use in characterizing its statistical uncertainty. Several approximate expressions have been used previously in the literature, but there is no consensus as to which is best. This paper compares three methods from the literature with a standard based on bootstrapping. Comparative calculations were carried out for nine major ions measured at 222 sampling sites in the National Atmospheric Deposition/National Trends Network (NADP/NTN). The ratio variance approximation of Cochran (1977) gave results that were not statistically different from those of bootstrapping, and is suggested as the method of choice for routine computing of the standard error of the weighted mean. The bootstrap method has advantages of its own, including the fact that it is nonparametric, but requires additional effort and computation time.}} The analytical formula which is the closest to the bootstrap is this one: var.wtd.mean.cochran <- function(x,w) # # Computes the variance of a weighted mean following Cochran 1977 definition # { n = length(w) xWbar = wtd.mean(x,w) wbar = mean(w) out = n/((n-1)*sum(w)^2)*(sum((w*x-wbar*xWbar)^2)-2*xWbar*sum((w- wbar)*(w*x-wbar*xWbar))+xWbar^2*sum((w-wbar)^2)) return(out) } It's the one I am retaining. NB: the part two of the paper cited above may also interest you: @article{Gatz1995AEa, Author = {Gatz, Donald F and Smith, Luther}, Date-Added = {2007-05-19 14:15:58 +0200}, Date-Modified = {2007-06-01 12:11:53 +0200}, Filed = {Yes}, Journal = {Atmospheric Environment}, Number = {11}, Pages = {1195--1200}, Title = {The standard error of a weighted mean concentration--II. Estimating confidence intervals}, Url = {http://www.sciencedirect.com/science/article/B6VH3-3YGV47C-6Y/ 2/a187487377ef52b741e3dabdfca97517}, Volume = {29}, Year = {1995}, Abstract = {One motivation for estimating the standard error, SEMw, of a weighted mean concentration, Mw, of an ion in precipitation is to use it to compute a confidence interval for Mw. Typically this is done by multiplying the standard error by a factor that depends on the degree of confidence one wishes to express, on the assumption that the weighted mean has a normal distribution. This paper compares confidence intervals of Mw concentrations of ions in precipitation, as computed using the assumption of a normal distribution, with those estimated from distributions produced by bootstrapping. The hypothesis that Mw was normally distributed was rejected about half the time (at the 5{\%} significance level) in tests involving nine major ions measured at ten diverse sites in the National Atmospheric Deposition Program/National Trends Network (NADP/NTN). Most of these rejections occurred at sites with fewer than 100 samples, in agreement with previous results. Nevertheless, the hypothesis was often rejected at sites with more than 100 samples as well. The maximum error (relative to Mw) in the 95{\%} confidence limits made by assuming a normal distribution of the Mw at the ten sites examined was about 27{\%}. Most such errors were less than 10{\%}, and errors were smaller at sampling sites with > 100 samples than at those with < 100 samples.}} Cheers, JiHO --- http://jo.irisson.free.fr/ ______________________________________________ [hidden email] mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. |
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